LMFDB - Embedded newform 6422.2.a.bn.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 4 x^{13} - 22 x^{12} + 98 x^{11} + 164 x^{10} - 912 x^{9} - 374 x^{8} + 3996 x^{7} - 817 x^{6} + \cdots + 358 $ x^14 - 4x^13 - 22x^12 + 98x^11 + 164x^10 - 912x^9 - 374x^8 + 3996x^7 - 817x^6 - 8194x^5 + 4418x^4 + 6430x^3 - 4327x^2 - 922*x + 358
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.50159$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.50159 q^{3} +1.00000 q^{4} +2.98870 q^{5} +1.50159 q^{6} +3.73146 q^{7} +1.00000 q^{8} -0.745218 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.50159 q^{3} +1.00000 q^{4} +2.98870 q^{5} +1.50159 q^{6} +3.73146 q^{7} +1.00000 q^{8} -0.745218 q^{9} +2.98870 q^{10} +3.81437 q^{11} +1.50159 q^{12} +3.73146 q^{14} +4.48781 q^{15} +1.00000 q^{16} -2.89129 q^{17} -0.745218 q^{18} +1.00000 q^{19} +2.98870 q^{20} +5.60313 q^{21} +3.81437 q^{22} -1.66948 q^{23} +1.50159 q^{24} +3.93232 q^{25} -5.62379 q^{27} +3.73146 q^{28} -8.51244 q^{29} +4.48781 q^{30} +1.71712 q^{31} +1.00000 q^{32} +5.72763 q^{33} -2.89129 q^{34} +11.1522 q^{35} -0.745218 q^{36} +5.22055 q^{37} +1.00000 q^{38} +2.98870 q^{40} +10.6438 q^{41} +5.60313 q^{42} +5.42719 q^{43} +3.81437 q^{44} -2.22723 q^{45} -1.66948 q^{46} -6.13466 q^{47} +1.50159 q^{48} +6.92379 q^{49} +3.93232 q^{50} -4.34155 q^{51} -13.2870 q^{53} -5.62379 q^{54} +11.4000 q^{55} +3.73146 q^{56} +1.50159 q^{57} -8.51244 q^{58} +5.52632 q^{59} +4.48781 q^{60} -5.99153 q^{61} +1.71712 q^{62} -2.78075 q^{63} +1.00000 q^{64} +5.72763 q^{66} -7.35059 q^{67} -2.89129 q^{68} -2.50689 q^{69} +11.1522 q^{70} -9.75998 q^{71} -0.745218 q^{72} -0.118233 q^{73} +5.22055 q^{74} +5.90474 q^{75} +1.00000 q^{76} +14.2332 q^{77} +3.04474 q^{79} +2.98870 q^{80} -6.20900 q^{81} +10.6438 q^{82} -1.22342 q^{83} +5.60313 q^{84} -8.64121 q^{85} +5.42719 q^{86} -12.7822 q^{87} +3.81437 q^{88} +14.8184 q^{89} -2.22723 q^{90} -1.66948 q^{92} +2.57842 q^{93} -6.13466 q^{94} +2.98870 q^{95} +1.50159 q^{96} +14.8790 q^{97} +6.92379 q^{98} -2.84253 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9}+O(q^{10}) $ 14 * q + 14 * q^2 + 4 * q^3 + 14 * q^4 - 2 * q^5 + 4 * q^6 + 2 * q^7 + 14 * q^8 + 18 * q^9	\( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9} - 2 q^{10} + 10 q^{11} + 4 q^{12} + 2 q^{14} - 4 q^{15} + 14 q^{16} + 2 q^{17} + 18 q^{18} + 14 q^{19} - 2 q^{20} - 18 q^{21} + 10 q^{22} + 12 q^{23} + 4 q^{24} + 44 q^{25} + 10 q^{27} + 2 q^{28} + 4 q^{29} - 4 q^{30} + 4 q^{31} + 14 q^{32} + 12 q^{33} + 2 q^{34} + 14 q^{35} + 18 q^{36} + 18 q^{37} + 14 q^{38} - 2 q^{40} + 6 q^{41} - 18 q^{42} + 28 q^{43} + 10 q^{44} + 8 q^{45} + 12 q^{46} - 20 q^{47} + 4 q^{48} + 28 q^{49} + 44 q^{50} + 10 q^{51} + 12 q^{53} + 10 q^{54} - 2 q^{55} + 2 q^{56} + 4 q^{57} + 4 q^{58} + 16 q^{59} - 4 q^{60} + 30 q^{61} + 4 q^{62} + 28 q^{63} + 14 q^{64} + 12 q^{66} + 2 q^{67} + 2 q^{68} - 42 q^{69} + 14 q^{70} + 44 q^{71} + 18 q^{72} - 36 q^{73} + 18 q^{74} + 46 q^{75} + 14 q^{76} + 68 q^{77} + 34 q^{79} - 2 q^{80} - 6 q^{81} + 6 q^{82} + 2 q^{83} - 18 q^{84} - 30 q^{85} + 28 q^{86} + 52 q^{87} + 10 q^{88} + 42 q^{89} + 8 q^{90} + 12 q^{92} - 12 q^{93} - 20 q^{94} - 2 q^{95} + 4 q^{96} + 40 q^{97} + 28 q^{98} + 12 q^{99}+O(q^{100}) \) 14 * q + 14 * q^2 + 4 * q^3 + 14 * q^4 - 2 * q^5 + 4 * q^6 + 2 * q^7 + 14 * q^8 + 18 * q^9 - 2 * q^10 + 10 * q^11 + 4 * q^12 + 2 * q^14 - 4 * q^15 + 14 * q^16 + 2 * q^17 + 18 * q^18 + 14 * q^19 - 2 * q^20 - 18 * q^21 + 10 * q^22 + 12 * q^23 + 4 * q^24 + 44 * q^25 + 10 * q^27 + 2 * q^28 + 4 * q^29 - 4 * q^30 + 4 * q^31 + 14 * q^32 + 12 * q^33 + 2 * q^34 + 14 * q^35 + 18 * q^36 + 18 * q^37 + 14 * q^38 - 2 * q^40 + 6 * q^41 - 18 * q^42 + 28 * q^43 + 10 * q^44 + 8 * q^45 + 12 * q^46 - 20 * q^47 + 4 * q^48 + 28 * q^49 + 44 * q^50 + 10 * q^51 + 12 * q^53 + 10 * q^54 - 2 * q^55 + 2 * q^56 + 4 * q^57 + 4 * q^58 + 16 * q^59 - 4 * q^60 + 30 * q^61 + 4 * q^62 + 28 * q^63 + 14 * q^64 + 12 * q^66 + 2 * q^67 + 2 * q^68 - 42 * q^69 + 14 * q^70 + 44 * q^71 + 18 * q^72 - 36 * q^73 + 18 * q^74 + 46 * q^75 + 14 * q^76 + 68 * q^77 + 34 * q^79 - 2 * q^80 - 6 * q^81 + 6 * q^82 + 2 * q^83 - 18 * q^84 - 30 * q^85 + 28 * q^86 + 52 * q^87 + 10 * q^88 + 42 * q^89 + 8 * q^90 + 12 * q^92 - 12 * q^93 - 20 * q^94 - 2 * q^95 + 4 * q^96 + 40 * q^97 + 28 * q^98 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.50159	0.866945	0.433473	−	0.901167i	$-0.357288\pi$
$3$	1.50159	0.866945	0.433473	+	0.901167i	$0.357288\pi$
$4$	1.00000	0.500000
$5$	2.98870	1.33659	0.668293	−	0.743898i	$-0.267025\pi$
$5$	2.98870	1.33659	0.668293	+	0.743898i	$0.267025\pi$
$6$	1.50159	0.613023
$7$	3.73146	1.41036	0.705180	−	0.709029i	$-0.250866\pi$
$7$	3.73146	1.41036	0.705180	+	0.709029i	$0.250866\pi$
$8$	1.00000	0.353553
$9$	−0.745218	−0.248406
$10$	2.98870	0.945109
$11$	3.81437	1.15007	0.575037	−	0.818127i	$-0.304988\pi$
$11$	3.81437	1.15007	0.575037	+	0.818127i	$0.304988\pi$
$12$	1.50159	0.433473
$13$	0	0
$13$	0	0
$14$	3.73146	0.997274
$15$	4.48781	1.15875
$16$	1.00000	0.250000
$17$	−2.89129	−0.701242	−0.350621	−	0.936517i	$-0.614029\pi$
$17$	−2.89129	−0.701242	−0.350621	+	0.936517i	$0.614029\pi$
$18$	−0.745218	−0.175650
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.98870	0.668293
$21$	5.60313	1.22270
$22$	3.81437	0.813225
$23$	−1.66948	−0.348112	−0.174056	−	0.984736i	$-0.555687\pi$
$23$	−1.66948	−0.348112	−0.174056	+	0.984736i	$0.555687\pi$
$24$	1.50159	0.306511
$25$	3.93232	0.786464
$26$	0	0
$27$	−5.62379	−1.08230
$28$	3.73146	0.705180
$29$	−8.51244	−1.58072	−0.790360	−	0.612643i	$-0.790106\pi$
$29$	−8.51244	−1.58072	−0.790360	+	0.612643i	$0.790106\pi$
$30$	4.48781	0.819358
$31$	1.71712	0.308404	0.154202	−	0.988039i	$-0.450719\pi$
$31$	1.71712	0.308404	0.154202	+	0.988039i	$0.450719\pi$
$32$	1.00000	0.176777
$33$	5.72763	0.997052
$34$	−2.89129	−0.495853
$35$	11.1522	1.88507
$36$	−0.745218	−0.124203
$37$	5.22055	0.858254	0.429127	−	0.903244i	$-0.358821\pi$
$37$	5.22055	0.858254	0.429127	+	0.903244i	$0.358821\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	2.98870	0.472555
$41$	10.6438	1.66229	0.831145	−	0.556056i	$-0.187686\pi$
$41$	10.6438	1.66229	0.831145	+	0.556056i	$0.187686\pi$
$42$	5.60313	0.864582
$43$	5.42719	0.827639	0.413820	−	0.910359i	$-0.364194\pi$
$43$	5.42719	0.827639	0.413820	+	0.910359i	$0.364194\pi$
$44$	3.81437	0.575037
$45$	−2.22723	−0.332016
$46$	−1.66948	−0.246152
$47$	−6.13466	−0.894831	−0.447416	−	0.894326i	$-0.647656\pi$
$47$	−6.13466	−0.894831	−0.447416	+	0.894326i	$0.647656\pi$
$48$	1.50159	0.216736
$49$	6.92379	0.989113
$50$	3.93232	0.556114
$51$	−4.34155	−0.607938
$52$	0	0
$53$	−13.2870	−1.82511	−0.912553	−	0.408959i	$-0.865892\pi$
$53$	−13.2870	−1.82511	−0.912553	+	0.408959i	$0.865892\pi$
$54$	−5.62379	−0.765301
$55$	11.4000	1.53717
$56$	3.73146	0.498637
$57$	1.50159	0.198891
$58$	−8.51244	−1.11774
$59$	5.52632	0.719465	0.359732	−	0.933055i	$-0.382868\pi$
$59$	5.52632	0.719465	0.359732	+	0.933055i	$0.382868\pi$
$60$	4.48781	0.579374
$61$	−5.99153	−0.767137	−0.383568	−	0.923512i	$-0.625305\pi$
$61$	−5.99153	−0.767137	−0.383568	+	0.923512i	$0.625305\pi$
$62$	1.71712	0.218075
$63$	−2.78075	−0.350342
$64$	1.00000	0.125000
$65$	0	0
$66$	5.72763	0.705022
$67$	−7.35059	−0.898018	−0.449009	−	0.893527i	$-0.648223\pi$
$67$	−7.35059	−0.898018	−0.449009	+	0.893527i	$0.648223\pi$
$68$	−2.89129	−0.350621
$69$	−2.50689	−0.301794
$70$	11.1522	1.33294
$71$	−9.75998	−1.15830	−0.579148	−	0.815222i	$-0.696615\pi$
$71$	−9.75998	−1.15830	−0.579148	+	0.815222i	$0.696615\pi$
$72$	−0.745218	−0.0878248
$73$	−0.118233	−0.0138382	−0.00691908	−	0.999976i	$-0.502202\pi$
$73$	−0.118233	−0.0138382	−0.00691908	+	0.999976i	$0.502202\pi$
$74$	5.22055	0.606877
$75$	5.90474	0.681821
$76$	1.00000	0.114708
$77$	14.2332	1.62202
$78$	0	0
$79$	3.04474	0.342560	0.171280	−	0.985222i	$-0.445210\pi$
$79$	3.04474	0.342560	0.171280	+	0.985222i	$0.445210\pi$
$80$	2.98870	0.334147
$81$	−6.20900	−0.689888
$82$	10.6438	1.17542
$83$	−1.22342	−0.134288	−0.0671439	−	0.997743i	$-0.521389\pi$
$83$	−1.22342	−0.134288	−0.0671439	+	0.997743i	$0.521389\pi$
$84$	5.60313	0.611352
$85$	−8.64121	−0.937271
$86$	5.42719	0.585229
$87$	−12.7822	−1.37040
$88$	3.81437	0.406613
$89$	14.8184	1.57074	0.785372	−	0.619023i	$-0.212471\pi$
$89$	14.8184	1.57074	0.785372	+	0.619023i	$0.212471\pi$
$90$	−2.22723	−0.234771
$91$	0	0
$92$	−1.66948	−0.174056
$93$	2.57842	0.267370
$94$	−6.13466	−0.632741
$95$	2.98870	0.306634
$96$	1.50159	0.153256
$97$	14.8790	1.51073	0.755366	−	0.655303i	$-0.227459\pi$
$97$	14.8790	1.51073	0.755366	+	0.655303i	$0.227459\pi$
$98$	6.92379	0.699408
$99$	−2.84253	−0.285685
$100$	3.93232	0.393232
$101$	14.4250	1.43534	0.717669	−	0.696384i	$-0.245209\pi$
$101$	14.4250	1.43534	0.717669	+	0.696384i	$0.245209\pi$
$102$	−4.34155	−0.429877
$103$	9.25772	0.912190	0.456095	−	0.889931i	$-0.349248\pi$
$103$	9.25772	0.912190	0.456095	+	0.889931i	$0.349248\pi$
$104$	0	0
$105$	16.7461	1.63425
$106$	−13.2870	−1.29054
$107$	−11.6631	−1.12752	−0.563759	−	0.825939i	$-0.690645\pi$
$107$	−11.6631	−1.12752	−0.563759	+	0.825939i	$0.690645\pi$
$108$	−5.62379	−0.541150
$109$	6.07064	0.581462	0.290731	−	0.956805i	$-0.406102\pi$
$109$	6.07064	0.581462	0.290731	+	0.956805i	$0.406102\pi$
$110$	11.4000	1.08695
$111$	7.83915	0.744059
$112$	3.73146	0.352590
$113$	−5.29092	−0.497728	−0.248864	−	0.968538i	$-0.580057\pi$
$113$	−5.29092	−0.497728	−0.248864	+	0.968538i	$0.580057\pi$
$114$	1.50159	0.140637
$115$	−4.98959	−0.465281
$116$	−8.51244	−0.790360
$117$	0	0
$118$	5.52632	0.508739
$119$	−10.7887	−0.989003
$120$	4.48781	0.409679
$121$	3.54939	0.322671
$122$	−5.99153	−0.542448
$123$	15.9827	1.44111
$124$	1.71712	0.154202
$125$	−3.19098	−0.285410
$126$	−2.78075	−0.247729
$127$	−11.2686	−0.999927	−0.499963	−	0.866046i	$-0.666653\pi$
$127$	−11.2686	−0.999927	−0.499963	+	0.866046i	$0.666653\pi$
$128$	1.00000	0.0883883
$129$	8.14944	0.717518
$130$	0	0
$131$	−12.7617	−1.11499	−0.557495	−	0.830180i	$-0.688238\pi$
$131$	−12.7617	−1.11499	−0.557495	+	0.830180i	$0.688238\pi$
$132$	5.72763	0.498526
$133$	3.73146	0.323559
$134$	−7.35059	−0.634994
$135$	−16.8078	−1.44659
$136$	−2.89129	−0.247926
$137$	−8.10390	−0.692363	−0.346182	−	0.938168i	$-0.612522\pi$
$137$	−8.10390	−0.692363	−0.346182	+	0.938168i	$0.612522\pi$
$138$	−2.50689	−0.213400
$139$	9.90009	0.839714	0.419857	−	0.907590i	$-0.362080\pi$
$139$	9.90009	0.839714	0.419857	+	0.907590i	$0.362080\pi$
$140$	11.1522	0.942534
$141$	−9.21176	−0.775770
$142$	−9.75998	−0.819039
$143$	0	0
$144$	−0.745218	−0.0621015
$145$	−25.4411	−2.11277
$146$	−0.118233	−0.00978506
$147$	10.3967	0.857506
$148$	5.22055	0.429127
$149$	−23.1692	−1.89809	−0.949045	−	0.315139i	$-0.897949\pi$
$149$	−23.1692	−1.89809	−0.949045	+	0.315139i	$0.897949\pi$
$150$	5.90474	0.482120
$151$	−20.1390	−1.63889	−0.819446	−	0.573157i	$-0.805718\pi$
$151$	−20.1390	−1.63889	−0.819446	+	0.573157i	$0.805718\pi$
$152$	1.00000	0.0811107
$153$	2.15464	0.174193
$154$	14.2332	1.14694
$155$	5.13196	0.412209
$156$	0	0
$157$	−13.0900	−1.04470	−0.522349	−	0.852732i	$-0.674944\pi$
$157$	−13.0900	−1.04470	−0.522349	+	0.852732i	$0.674944\pi$
$158$	3.04474	0.242226
$159$	−19.9516	−1.58227
$160$	2.98870	0.236277
$161$	−6.22961	−0.490962
$162$	−6.20900	−0.487825
$163$	5.90255	0.462323	0.231162	−	0.972915i	$-0.425747\pi$
$163$	5.90255	0.462323	0.231162	+	0.972915i	$0.425747\pi$
$164$	10.6438	0.831145
$165$	17.1181	1.33265
$166$	−1.22342	−0.0949559
$167$	16.6567	1.28894	0.644468	−	0.764631i	$-0.277079\pi$
$167$	16.6567	1.28894	0.644468	+	0.764631i	$0.277079\pi$
$168$	5.60313	0.432291
$169$	0	0
$170$	−8.64121	−0.662750
$171$	−0.745218	−0.0569882
$172$	5.42719	0.413820
$173$	−9.47289	−0.720211	−0.360105	−	0.932912i	$-0.617259\pi$
$173$	−9.47289	−0.720211	−0.360105	+	0.932912i	$0.617259\pi$
$174$	−12.7822	−0.969017
$175$	14.6733	1.10920
$176$	3.81437	0.287519
$177$	8.29828	0.623737
$178$	14.8184	1.11068
$179$	−5.00441	−0.374047	−0.187024	−	0.982355i	$-0.559884\pi$
$179$	−5.00441	−0.374047	−0.187024	+	0.982355i	$0.559884\pi$
$180$	−2.22723	−0.166008
$181$	−1.97399	−0.146726	−0.0733629	−	0.997305i	$-0.523373\pi$
$181$	−1.97399	−0.146726	−0.0733629	+	0.997305i	$0.523373\pi$
$182$	0	0
$183$	−8.99684	−0.665065
$184$	−1.66948	−0.123076
$185$	15.6027	1.14713
$186$	2.57842	0.189059
$187$	−11.0285	−0.806480
$188$	−6.13466	−0.447416
$189$	−20.9850	−1.52643
$190$	2.98870	0.216823
$191$	−20.8136	−1.50602	−0.753010	−	0.658009i	$-0.771399\pi$
$191$	−20.8136	−1.50602	−0.753010	+	0.658009i	$0.771399\pi$
$192$	1.50159	0.108368
$193$	21.3064	1.53367	0.766833	−	0.641847i	$-0.221831\pi$
$193$	21.3064	1.53367	0.766833	+	0.641847i	$0.221831\pi$
$194$	14.8790	1.06825
$195$	0	0
$196$	6.92379	0.494556
$197$	−8.62246	−0.614325	−0.307162	−	0.951657i	$-0.599379\pi$
$197$	−8.62246	−0.614325	−0.307162	+	0.951657i	$0.599379\pi$
$198$	−2.84253	−0.202010
$199$	4.00411	0.283844	0.141922	−	0.989878i	$-0.454672\pi$
$199$	4.00411	0.283844	0.141922	+	0.989878i	$0.454672\pi$
$200$	3.93232	0.278057
$201$	−11.0376	−0.778532
$202$	14.4250	1.01494
$203$	−31.7638	−2.22938
$204$	−4.34155	−0.303969
$205$	31.8112	2.22179
$206$	9.25772	0.645016
$207$	1.24413	0.0864730
$208$	0	0
$209$	3.81437	0.263845
$210$	16.7461	1.15559
$211$	8.29916	0.571338	0.285669	−	0.958328i	$-0.407784\pi$
$211$	8.29916	0.571338	0.285669	+	0.958328i	$0.407784\pi$
$212$	−13.2870	−0.912553
$213$	−14.6555	−1.00418
$214$	−11.6631	−0.797276
$215$	16.2202	1.10621
$216$	−5.62379	−0.382651
$217$	6.40737	0.434961
$218$	6.07064	0.411155
$219$	−0.177538	−0.0119969
$220$	11.4000	0.768587
$221$	0	0
$222$	7.83915	0.526129
$223$	8.81599	0.590362	0.295181	−	0.955441i	$-0.404620\pi$
$223$	8.81599	0.590362	0.295181	+	0.955441i	$0.404620\pi$
$224$	3.73146	0.249319
$225$	−2.93043	−0.195362
$226$	−5.29092	−0.351947
$227$	−4.18895	−0.278030	−0.139015	−	0.990290i	$-0.544394\pi$
$227$	−4.18895	−0.278030	−0.139015	+	0.990290i	$0.544394\pi$
$228$	1.50159	0.0994454
$229$	−24.2635	−1.60338	−0.801689	−	0.597741i	$-0.796065\pi$
$229$	−24.2635	−1.60338	−0.801689	+	0.597741i	$0.796065\pi$
$230$	−4.98959	−0.329004
$231$	21.3724	1.40620
$232$	−8.51244	−0.558869
$233$	17.9212	1.17405	0.587027	−	0.809567i	$-0.300298\pi$
$233$	17.9212	1.17405	0.587027	+	0.809567i	$0.300298\pi$
$234$	0	0
$235$	−18.3346	−1.19602
$236$	5.52632	0.359732
$237$	4.57196	0.296981
$238$	−10.7887	−0.699331
$239$	−22.3178	−1.44362	−0.721808	−	0.692093i	$-0.756689\pi$
$239$	−22.3178	−1.44362	−0.721808	+	0.692093i	$0.756689\pi$
$240$	4.48781	0.289687
$241$	22.3962	1.44267	0.721334	−	0.692588i	$-0.243529\pi$
$241$	22.3962	1.44267	0.721334	+	0.692588i	$0.243529\pi$
$242$	3.54939	0.228163
$243$	7.54799	0.484204
$244$	−5.99153	−0.383568
$245$	20.6931	1.32203
$246$	15.9827	1.01902
$247$	0	0
$248$	1.71712	0.109037
$249$	−1.83708	−0.116420
$250$	−3.19098	−0.201815
$251$	3.68665	0.232699	0.116350	−	0.993208i	$-0.462881\pi$
$251$	3.68665	0.232699	0.116350	+	0.993208i	$0.462881\pi$
$252$	−2.78075	−0.175171
$253$	−6.36802	−0.400354
$254$	−11.2686	−0.707055
$255$	−12.9756	−0.812562
$256$	1.00000	0.0625000
$257$	12.7082	0.792718	0.396359	−	0.918096i	$-0.370274\pi$
$257$	12.7082	0.792718	0.396359	+	0.918096i	$0.370274\pi$
$258$	8.14944	0.507362
$259$	19.4803	1.21045
$260$	0	0
$261$	6.34362	0.392660
$262$	−12.7617	−0.788418
$263$	3.24916	0.200352	0.100176	−	0.994970i	$-0.468059\pi$
$263$	3.24916	0.200352	0.100176	+	0.994970i	$0.468059\pi$
$264$	5.72763	0.352511
$265$	−39.7107	−2.43941
$266$	3.73146	0.228790
$267$	22.2512	1.36175
$268$	−7.35059	−0.449009
$269$	10.9402	0.667037	0.333519	−	0.942744i	$-0.391764\pi$
$269$	10.9402	0.667037	0.333519	+	0.942744i	$0.391764\pi$
$270$	−16.8078	−1.02289
$271$	−0.968555	−0.0588355	−0.0294178	−	0.999567i	$-0.509365\pi$
$271$	−0.968555	−0.0588355	−0.0294178	+	0.999567i	$0.509365\pi$
$272$	−2.89129	−0.175310
$273$	0	0
$274$	−8.10390	−0.489575
$275$	14.9993	0.904492
$276$	−2.50689	−0.150897
$277$	8.02640	0.482260	0.241130	−	0.970493i	$-0.422482\pi$
$277$	8.02640	0.482260	0.241130	+	0.970493i	$0.422482\pi$
$278$	9.90009	0.593768
$279$	−1.27963	−0.0766094
$280$	11.1522	0.666472
$281$	−4.41792	−0.263551	−0.131775	−	0.991280i	$-0.542068\pi$
$281$	−4.41792	−0.263551	−0.131775	+	0.991280i	$0.542068\pi$
$282$	−9.21176	−0.548552
$283$	−15.7900	−0.938621	−0.469310	−	0.883033i	$-0.655497\pi$
$283$	−15.7900	−0.938621	−0.469310	+	0.883033i	$0.655497\pi$
$284$	−9.75998	−0.579148
$285$	4.48781	0.265835
$286$	0	0
$287$	39.7171	2.34442
$288$	−0.745218	−0.0439124
$289$	−8.64042	−0.508260
$290$	−25.4411	−1.49395
$291$	22.3422	1.30972
$292$	−0.118233	−0.00691908
$293$	34.0058	1.98664	0.993320	−	0.115389i	$-0.0368115\pi$
$293$	34.0058	1.98664	0.993320	+	0.115389i	$0.0368115\pi$
$294$	10.3967	0.606349
$295$	16.5165	0.961627
$296$	5.22055	0.303438
$297$	−21.4512	−1.24473
$298$	−23.1692	−1.34215
$299$	0	0
$300$	5.90474	0.340911
$301$	20.2513	1.16727
$302$	−20.1390	−1.15887
$303$	21.6604	1.24436
$304$	1.00000	0.0573539
$305$	−17.9069	−1.02534
$306$	2.15464	0.123173
$307$	−29.2156	−1.66743	−0.833713	−	0.552199i	$-0.813789\pi$
$307$	−29.2156	−1.66743	−0.833713	+	0.552199i	$0.813789\pi$
$308$	14.2332	0.811009
$309$	13.9013	0.790819
$310$	5.13196	0.291476
$311$	−13.1516	−0.745761	−0.372881	−	0.927879i	$-0.621630\pi$
$311$	−13.1516	−0.745761	−0.372881	+	0.927879i	$0.621630\pi$
$312$	0	0
$313$	22.1828	1.25385	0.626924	−	0.779080i	$-0.284314\pi$
$313$	22.1828	1.25385	0.626924	+	0.779080i	$0.284314\pi$
$314$	−13.0900	−0.738712
$315$	−8.31082	−0.468262
$316$	3.04474	0.171280
$317$	−25.2932	−1.42061	−0.710303	−	0.703896i	$-0.751442\pi$
$317$	−25.2932	−1.42061	−0.710303	+	0.703896i	$0.751442\pi$
$318$	−19.9516	−1.11883
$319$	−32.4695	−1.81795
$320$	2.98870	0.167073
$321$	−17.5133	−0.977496
$322$	−6.22961	−0.347163
$323$	−2.89129	−0.160876
$324$	−6.20900	−0.344944
$325$	0	0
$326$	5.90255	0.326912
$327$	9.11563	0.504095
$328$	10.6438	0.587708
$329$	−22.8912	−1.26203
$330$	17.1181	0.942323
$331$	27.6235	1.51832	0.759162	−	0.650901i	$-0.225609\pi$
$331$	27.6235	1.51832	0.759162	+	0.650901i	$0.225609\pi$
$332$	−1.22342	−0.0671439
$333$	−3.89045	−0.213195
$334$	16.6567	0.911416
$335$	−21.9687	−1.20028
$336$	5.60313	0.305676
$337$	−21.3984	−1.16565	−0.582824	−	0.812599i	$-0.698052\pi$
$337$	−21.3984	−1.16565	−0.582824	+	0.812599i	$0.698052\pi$
$338$	0	0
$339$	−7.94481	−0.431503
$340$	−8.64121	−0.468635
$341$	6.54973	0.354688
$342$	−0.745218	−0.0402968
$343$	−0.284381	−0.0153551
$344$	5.42719	0.292615
$345$	−7.49233	−0.403373
$346$	−9.47289	−0.509266
$347$	−8.49897	−0.456249	−0.228124	−	0.973632i	$-0.573259\pi$
$347$	−8.49897	−0.456249	−0.228124	+	0.973632i	$0.573259\pi$
$348$	−12.7822	−0.685199
$349$	−21.6315	−1.15791	−0.578953	−	0.815361i	$-0.696538\pi$
$349$	−21.6315	−1.15791	−0.578953	+	0.815361i	$0.696538\pi$
$350$	14.6733	0.784320
$351$	0	0
$352$	3.81437	0.203306
$353$	28.9686	1.54184	0.770922	−	0.636930i	$-0.219796\pi$
$353$	28.9686	1.54184	0.770922	+	0.636930i	$0.219796\pi$
$354$	8.29828	0.441048
$355$	−29.1696	−1.54816
$356$	14.8184	0.785372
$357$	−16.2003	−0.857411
$358$	−5.00441	−0.264491
$359$	7.42119	0.391675	0.195838	−	0.980636i	$-0.437257\pi$
$359$	7.42119	0.391675	0.195838	+	0.980636i	$0.437257\pi$
$360$	−2.22723	−0.117385
$361$	1.00000	0.0526316
$362$	−1.97399	−0.103751
$363$	5.32973	0.279738
$364$	0	0
$365$	−0.353364	−0.0184959
$366$	−8.99684	−0.470272
$367$	13.6135	0.710619	0.355310	−	0.934749i	$-0.384375\pi$
$367$	13.6135	0.710619	0.355310	+	0.934749i	$0.384375\pi$
$368$	−1.66948	−0.0870279
$369$	−7.93198	−0.412923
$370$	15.6027	0.811144
$371$	−49.5798	−2.57405
$372$	2.57842	0.133685
$373$	−2.73685	−0.141709	−0.0708544	−	0.997487i	$-0.522573\pi$
$373$	−2.73685	−0.141709	−0.0708544	+	0.997487i	$0.522573\pi$
$374$	−11.0285	−0.570268
$375$	−4.79155	−0.247434
$376$	−6.13466	−0.316371
$377$	0	0
$378$	−20.9850	−1.07935
$379$	−1.62208	−0.0833208	−0.0416604	−	0.999132i	$-0.513265\pi$
$379$	−1.62208	−0.0833208	−0.0416604	+	0.999132i	$0.513265\pi$
$380$	2.98870	0.153317
$381$	−16.9209	−0.866882
$382$	−20.8136	−1.06492
$383$	31.9773	1.63396	0.816981	−	0.576664i	$-0.195646\pi$
$383$	31.9773	1.63396	0.816981	+	0.576664i	$0.195646\pi$
$384$	1.50159	0.0766279
$385$	42.5386	2.16797
$386$	21.3064	1.08447
$387$	−4.04444	−0.205591
$388$	14.8790	0.755366
$389$	−4.42183	−0.224195	−0.112098	−	0.993697i	$-0.535757\pi$
$389$	−4.42183	−0.224195	−0.112098	+	0.993697i	$0.535757\pi$
$390$	0	0
$391$	4.82697	0.244110
$392$	6.92379	0.349704
$393$	−19.1628	−0.966636
$394$	−8.62246	−0.434393
$395$	9.09981	0.457861
$396$	−2.84253	−0.142843
$397$	0.227725	0.0114292	0.00571459	−	0.999984i	$-0.498181\pi$
$397$	0.227725	0.0114292	0.00571459	+	0.999984i	$0.498181\pi$
$398$	4.00411	0.200708
$399$	5.60313	0.280508
$400$	3.93232	0.196616
$401$	6.80150	0.339651	0.169825	−	0.985474i	$-0.445680\pi$
$401$	6.80150	0.339651	0.169825	+	0.985474i	$0.445680\pi$
$402$	−11.0376	−0.550505
$403$	0	0
$404$	14.4250	0.717669
$405$	−18.5568	−0.922096
$406$	−31.7638	−1.57641
$407$	19.9131	0.987056
$408$	−4.34155	−0.214939
$409$	16.0814	0.795175	0.397588	−	0.917564i	$-0.369847\pi$
$409$	16.0814	0.795175	0.397588	+	0.917564i	$0.369847\pi$
$410$	31.8112	1.57105
$411$	−12.1688	−0.600241
$412$	9.25772	0.456095
$413$	20.6212	1.01470
$414$	1.24413	0.0611456
$415$	−3.65644	−0.179487
$416$	0	0
$417$	14.8659	0.727986
$418$	3.81437	0.186567
$419$	−21.3995	−1.04544	−0.522718	−	0.852506i	$-0.675082\pi$
$419$	−21.3995	−1.04544	−0.522718	+	0.852506i	$0.675082\pi$
$420$	16.7461	0.817125
$421$	26.3895	1.28615	0.643074	−	0.765804i	$-0.277659\pi$
$421$	26.3895	1.28615	0.643074	+	0.765804i	$0.277659\pi$
$422$	8.29916	0.403997
$423$	4.57165	0.222281
$424$	−13.2870	−0.645272
$425$	−11.3695	−0.551501
$426$	−14.6555	−0.710062
$427$	−22.3571	−1.08194
$428$	−11.6631	−0.563759
$429$	0	0
$430$	16.2202	0.782210
$431$	22.5479	1.08609	0.543047	−	0.839702i	$-0.317271\pi$
$431$	22.5479	1.08609	0.543047	+	0.839702i	$0.317271\pi$
$432$	−5.62379	−0.270575
$433$	11.5129	0.553273	0.276636	−	0.960975i	$-0.410780\pi$
$433$	11.5129	0.553273	0.276636	+	0.960975i	$0.410780\pi$
$434$	6.40737	0.307564
$435$	−38.2022	−1.83165
$436$	6.07064	0.290731
$437$	−1.66948	−0.0798623
$438$	−0.177538	−0.00848311
$439$	−11.6448	−0.555778	−0.277889	−	0.960613i	$-0.589635\pi$
$439$	−11.6448	−0.555778	−0.277889	+	0.960613i	$0.589635\pi$
$440$	11.4000	0.543473
$441$	−5.15973	−0.245701
$442$	0	0
$443$	26.0168	1.23609	0.618047	−	0.786141i	$-0.287924\pi$
$443$	26.0168	1.23609	0.618047	+	0.786141i	$0.287924\pi$
$444$	7.83915	0.372029
$445$	44.2877	2.09944
$446$	8.81599	0.417449
$447$	−34.7906	−1.64554
$448$	3.73146	0.176295
$449$	11.7314	0.553640	0.276820	−	0.960922i	$-0.410719\pi$
$449$	11.7314	0.553640	0.276820	+	0.960922i	$0.410719\pi$
$450$	−2.93043	−0.138142
$451$	40.5995	1.91176
$452$	−5.29092	−0.248864
$453$	−30.2406	−1.42083
$454$	−4.18895	−0.196597
$455$	0	0
$456$	1.50159	0.0703185
$457$	−36.7678	−1.71993	−0.859963	−	0.510356i	$-0.829514\pi$
$457$	−36.7678	−1.71993	−0.859963	+	0.510356i	$0.829514\pi$
$458$	−24.2635	−1.13376
$459$	16.2600	0.758954
$460$	−4.98959	−0.232641
$461$	−12.0022	−0.558996	−0.279498	−	0.960146i	$-0.590168\pi$
$461$	−12.0022	−0.558996	−0.279498	+	0.960146i	$0.590168\pi$
$462$	21.3724	0.994334
$463$	15.5101	0.720815	0.360407	−	0.932795i	$-0.382638\pi$
$463$	15.5101	0.720815	0.360407	+	0.932795i	$0.382638\pi$
$464$	−8.51244	−0.395180
$465$	7.70612	0.357363
$466$	17.9212	0.830182
$467$	−25.8302	−1.19528	−0.597639	−	0.801765i	$-0.703894\pi$
$467$	−25.8302	−1.19528	−0.597639	+	0.801765i	$0.703894\pi$
$468$	0	0
$469$	−27.4284	−1.26653
$470$	−18.3346	−0.845714
$471$	−19.6559	−0.905695
$472$	5.52632	0.254369
$473$	20.7013	0.951847
$474$	4.57196	0.209997
$475$	3.93232	0.180427
$476$	−10.7887	−0.494501
$477$	9.90169	0.453367
$478$	−22.3178	−1.02079
$479$	10.0622	0.459753	0.229876	−	0.973220i	$-0.426168\pi$
$479$	10.0622	0.459753	0.229876	+	0.973220i	$0.426168\pi$
$480$	4.48781	0.204840
$481$	0	0
$482$	22.3962	1.02012
$483$	−9.35434	−0.425637
$484$	3.54939	0.161336
$485$	44.4688	2.01922
$486$	7.54799	0.342384
$487$	−5.76723	−0.261338	−0.130669	−	0.991426i	$-0.541713\pi$
$487$	−5.76723	−0.261338	−0.130669	+	0.991426i	$0.541713\pi$
$488$	−5.99153	−0.271224
$489$	8.86323	0.400809
$490$	20.6931	0.934820
$491$	19.4673	0.878546	0.439273	−	0.898354i	$-0.355236\pi$
$491$	19.4673	0.878546	0.439273	+	0.898354i	$0.355236\pi$
$492$	15.9827	0.720557
$493$	24.6120	1.10847
$494$	0	0
$495$	−8.49548	−0.381843
$496$	1.71712	0.0771010
$497$	−36.4190	−1.63361
$498$	−1.83708	−0.0823215
$499$	4.74563	0.212444	0.106222	−	0.994342i	$-0.466125\pi$
$499$	4.74563	0.212444	0.106222	+	0.994342i	$0.466125\pi$
$500$	−3.19098	−0.142705
$501$	25.0116	1.11744
$502$	3.68665	0.164543
$503$	−30.4469	−1.35756	−0.678779	−	0.734342i	$-0.737491\pi$
$503$	−30.4469	−1.35756	−0.678779	+	0.734342i	$0.737491\pi$
$504$	−2.78075	−0.123864
$505$	43.1119	1.91845
$506$	−6.36802	−0.283093
$507$	0	0
$508$	−11.2686	−0.499963
$509$	−6.35941	−0.281876	−0.140938	−	0.990018i	$-0.545012\pi$
$509$	−6.35941	−0.281876	−0.140938	+	0.990018i	$0.545012\pi$
$510$	−12.9756	−0.574568
$511$	−0.441183	−0.0195168
$512$	1.00000	0.0441942
$513$	−5.62379	−0.248297
$514$	12.7082	0.560536
$515$	27.6685	1.21922
$516$	8.14944	0.358759
$517$	−23.3998	−1.02912
$518$	19.4803	0.855914
$519$	−14.2244	−0.624383
$520$	0	0
$521$	8.30076	0.363663	0.181832	−	0.983330i	$-0.441797\pi$
$521$	8.30076	0.363663	0.181832	+	0.983330i	$0.441797\pi$
$522$	6.34362	0.277653
$523$	0.111917	0.00489377	0.00244688	−	0.999997i	$-0.499221\pi$
$523$	0.111917	0.00489377	0.00244688	+	0.999997i	$0.499221\pi$
$524$	−12.7617	−0.557495
$525$	22.0333	0.961613
$526$	3.24916	0.141670
$527$	−4.96470	−0.216266
$528$	5.72763	0.249263
$529$	−20.2128	−0.878818
$530$	−39.7107	−1.72492
$531$	−4.11831	−0.178719
$532$	3.73146	0.161779
$533$	0	0
$534$	22.2512	0.962903
$535$	−34.8576	−1.50703
$536$	−7.35059	−0.317497
$537$	−7.51459	−0.324278
$538$	10.9402	0.471667
$539$	26.4099	1.13755
$540$	−16.8078	−0.723294
$541$	36.1474	1.55410	0.777049	−	0.629440i	$-0.216716\pi$
$541$	36.1474	1.55410	0.777049	+	0.629440i	$0.216716\pi$
$542$	−0.968555	−0.0416030
$543$	−2.96414	−0.127203
$544$	−2.89129	−0.123963
$545$	18.1433	0.777174
$546$	0	0
$547$	−5.56274	−0.237846	−0.118923	−	0.992904i	$-0.537944\pi$
$547$	−5.56274	−0.237846	−0.118923	+	0.992904i	$0.537944\pi$
$548$	−8.10390	−0.346182
$549$	4.46500	0.190561
$550$	14.9993	0.639572
$551$	−8.51244	−0.362642
$552$	−2.50689	−0.106700
$553$	11.3613	0.483132
$554$	8.02640	0.341009
$555$	23.4288	0.994499
$556$	9.90009	0.419857
$557$	12.7166	0.538820	0.269410	−	0.963026i	$-0.413171\pi$
$557$	12.7166	0.538820	0.269410	+	0.963026i	$0.413171\pi$
$558$	−1.27963	−0.0541711
$559$	0	0
$560$	11.1522	0.471267
$561$	−16.5603	−0.699174
$562$	−4.41792	−0.186359
$563$	−32.2272	−1.35821	−0.679106	−	0.734040i	$-0.737633\pi$
$563$	−32.2272	−1.35821	−0.679106	+	0.734040i	$0.737633\pi$
$564$	−9.21176	−0.387885
$565$	−15.8130	−0.665256
$566$	−15.7900	−0.663705
$567$	−23.1686	−0.972990
$568$	−9.75998	−0.409520
$569$	−3.69103	−0.154736	−0.0773680	−	0.997003i	$-0.524652\pi$
$569$	−3.69103	−0.154736	−0.0773680	+	0.997003i	$0.524652\pi$
$570$	4.48781	0.187974
$571$	−21.2298	−0.888440	−0.444220	−	0.895918i	$-0.646519\pi$
$571$	−21.2298	−0.888440	−0.444220	+	0.895918i	$0.646519\pi$
$572$	0	0
$573$	−31.2536	−1.30564
$574$	39.7171	1.65776
$575$	−6.56495	−0.273777
$576$	−0.745218	−0.0310507
$577$	16.7181	0.695983	0.347992	−	0.937498i	$-0.386864\pi$
$577$	16.7181	0.695983	0.347992	+	0.937498i	$0.386864\pi$
$578$	−8.64042	−0.359394
$579$	31.9935	1.32960
$580$	−25.4411	−1.05638
$581$	−4.56514	−0.189394
$582$	22.3422	0.926113
$583$	−50.6814	−2.09901
$584$	−0.118233	−0.00489253
$585$	0	0
$586$	34.0058	1.40477
$587$	25.7669	1.06351	0.531757	−	0.846897i	$-0.321532\pi$
$587$	25.7669	1.06351	0.531757	+	0.846897i	$0.321532\pi$
$588$	10.3967	0.428753
$589$	1.71712	0.0707528
$590$	16.5165	0.679973
$591$	−12.9474	−0.532586
$592$	5.22055	0.214563
$593$	−22.0804	−0.906734	−0.453367	−	0.891324i	$-0.649777\pi$
$593$	−22.0804	−0.906734	−0.453367	+	0.891324i	$0.649777\pi$
$594$	−21.4512	−0.880154
$595$	−32.2443	−1.32189
$596$	−23.1692	−0.949045
$597$	6.01254	0.246077
$598$	0	0
$599$	−24.2400	−0.990421	−0.495211	−	0.868773i	$-0.664909\pi$
$599$	−24.2400	−0.990421	−0.495211	+	0.868773i	$0.664909\pi$
$600$	5.90474	0.241060
$601$	−25.0975	−1.02375	−0.511875	−	0.859060i	$-0.671049\pi$
$601$	−25.0975	−1.02375	−0.511875	+	0.859060i	$0.671049\pi$
$602$	20.2513	0.825383
$603$	5.47779	0.223073
$604$	−20.1390	−0.819446
$605$	10.6080	0.431278
$606$	21.6604	0.879895
$607$	39.1982	1.59101	0.795503	−	0.605950i	$-0.207207\pi$
$607$	39.1982	1.59101	0.795503	+	0.605950i	$0.207207\pi$
$608$	1.00000	0.0405554
$609$	−47.6963	−1.93275
$610$	−17.9069	−0.725028
$611$	0	0
$612$	2.15464	0.0870963
$613$	−31.8846	−1.28781	−0.643904	−	0.765106i	$-0.722686\pi$
$613$	−31.8846	−1.28781	−0.643904	+	0.765106i	$0.722686\pi$
$614$	−29.2156	−1.17905
$615$	47.7675	1.92617
$616$	14.2332	0.573470
$617$	29.2084	1.17589	0.587944	−	0.808902i	$-0.299938\pi$
$617$	29.2084	1.17589	0.587944	+	0.808902i	$0.299938\pi$
$618$	13.9013	0.559193
$619$	34.6290	1.39186	0.695929	−	0.718111i	$-0.254993\pi$
$619$	34.6290	1.39186	0.695929	+	0.718111i	$0.254993\pi$
$620$	5.13196	0.206104
$621$	9.38884	0.376761
$622$	−13.1516	−0.527333
$623$	55.2942	2.21531
$624$	0	0
$625$	−29.1985	−1.16794
$626$	22.1828	0.886605
$627$	5.72763	0.228739
$628$	−13.0900	−0.522349
$629$	−15.0942	−0.601843
$630$	−8.31082	−0.331111
$631$	−16.9719	−0.675640	−0.337820	−	0.941211i	$-0.609689\pi$
$631$	−16.9719	−0.675640	−0.337820	+	0.941211i	$0.609689\pi$
$632$	3.04474	0.121113
$633$	12.4620	0.495318
$634$	−25.2932	−1.00452
$635$	−33.6785	−1.33649
$636$	−19.9516	−0.791133
$637$	0	0
$638$	−32.4695	−1.28548
$639$	7.27331	0.287728
$640$	2.98870	0.118139
$641$	−22.2794	−0.879982	−0.439991	−	0.898002i	$-0.645018\pi$
$641$	−22.2794	−0.879982	−0.439991	+	0.898002i	$0.645018\pi$
$642$	−17.5133	−0.691194
$643$	−42.5539	−1.67816	−0.839080	−	0.544007i	$-0.816906\pi$
$643$	−42.5539	−1.67816	−0.839080	+	0.544007i	$0.816906\pi$
$644$	−6.22961	−0.245481
$645$	24.3562	0.959025
$646$	−2.89129	−0.113756
$647$	37.1643	1.46108	0.730539	−	0.682871i	$-0.239269\pi$
$647$	37.1643	1.46108	0.730539	+	0.682871i	$0.239269\pi$
$648$	−6.20900	−0.243912
$649$	21.0794	0.827438
$650$	0	0
$651$	9.62126	0.377087
$652$	5.90255	0.231162
$653$	−42.4412	−1.66085	−0.830427	−	0.557127i	$-0.811904\pi$
$653$	−42.4412	−1.66085	−0.830427	+	0.557127i	$0.811904\pi$
$654$	9.11563	0.356449
$655$	−38.1407	−1.49028
$656$	10.6438	0.415572
$657$	0.0881096	0.00343748
$658$	−22.8912	−0.892392
$659$	−32.8747	−1.28062	−0.640308	−	0.768118i	$-0.721193\pi$
$659$	−32.8747	−1.28062	−0.640308	+	0.768118i	$0.721193\pi$
$660$	17.1181	0.666323
$661$	15.2765	0.594189	0.297094	−	0.954848i	$-0.403982\pi$
$661$	15.2765	0.594189	0.297094	+	0.954848i	$0.403982\pi$
$662$	27.6235	1.07362
$663$	0	0
$664$	−1.22342	−0.0474779
$665$	11.1522	0.432464
$666$	−3.89045	−0.150752
$667$	14.2114	0.550267
$668$	16.6567	0.644468
$669$	13.2380	0.511812
$670$	−21.9687	−0.848725
$671$	−22.8539	−0.882264
$672$	5.60313	0.216146
$673$	6.81905	0.262855	0.131428	−	0.991326i	$-0.458044\pi$
$673$	6.81905	0.262855	0.131428	+	0.991326i	$0.458044\pi$
$674$	−21.3984	−0.824237
$675$	−22.1146	−0.851190
$676$	0	0
$677$	21.0962	0.810793	0.405396	−	0.914141i	$-0.367134\pi$
$677$	21.0962	0.810793	0.405396	+	0.914141i	$0.367134\pi$
$678$	−7.94481	−0.305119
$679$	55.5203	2.13067
$680$	−8.64121	−0.331375
$681$	−6.29010	−0.241037
$682$	6.54973	0.250802
$683$	43.7151	1.67271	0.836355	−	0.548188i	$-0.184682\pi$
$683$	43.7151	1.67271	0.836355	+	0.548188i	$0.184682\pi$
$684$	−0.745218	−0.0284941
$685$	−24.2201	−0.925403
$686$	−0.284381	−0.0108577
$687$	−36.4339	−1.39004
$688$	5.42719	0.206910
$689$	0	0
$690$	−7.49233	−0.285228
$691$	43.6813	1.66172	0.830858	−	0.556485i	$-0.187850\pi$
$691$	43.6813	1.66172	0.830858	+	0.556485i	$0.187850\pi$
$692$	−9.47289	−0.360105
$693$	−10.6068	−0.402919
$694$	−8.49897	−0.322617
$695$	29.5884	1.12235
$696$	−12.7822	−0.484509
$697$	−30.7745	−1.16567
$698$	−21.6315	−0.818763
$699$	26.9103	1.01784
$700$	14.6733	0.554598
$701$	12.9613	0.489540	0.244770	−	0.969581i	$-0.421288\pi$
$701$	12.9613	0.489540	0.244770	+	0.969581i	$0.421288\pi$
$702$	0	0
$703$	5.22055	0.196897
$704$	3.81437	0.143759
$705$	−27.5312	−1.03688
$706$	28.9686	1.09025
$707$	53.8262	2.02434
$708$	8.29828	0.311868
$709$	−3.19138	−0.119855	−0.0599275	−	0.998203i	$-0.519087\pi$
$709$	−3.19138	−0.119855	−0.0599275	+	0.998203i	$0.519087\pi$
$710$	−29.1696	−1.09472
$711$	−2.26899	−0.0850939
$712$	14.8184	0.555342
$713$	−2.86671	−0.107359
$714$	−16.2003	−0.606281
$715$	0	0
$716$	−5.00441	−0.187024
$717$	−33.5122	−1.25154
$718$	7.42119	0.276956
$719$	52.2035	1.94686	0.973431	−	0.228980i	$-0.0735391\pi$
$719$	52.2035	1.94686	0.973431	+	0.228980i	$0.0735391\pi$
$720$	−2.22723	−0.0830040
$721$	34.5448	1.28652
$722$	1.00000	0.0372161
$723$	33.6300	1.25071
$724$	−1.97399	−0.0733629
$725$	−33.4736	−1.24318
$726$	5.32973	0.197805
$727$	−44.7559	−1.65990	−0.829952	−	0.557835i	$-0.811632\pi$
$727$	−44.7559	−1.65990	−0.829952	+	0.557835i	$0.811632\pi$
$728$	0	0
$729$	29.9610	1.10967
$730$	−0.353364	−0.0130786
$731$	−15.6916	−0.580375
$732$	−8.99684	−0.332533
$733$	−12.8805	−0.475753	−0.237877	−	0.971295i	$-0.576451\pi$
$733$	−12.8805	−0.475753	−0.237877	+	0.971295i	$0.576451\pi$
$734$	13.6135	0.502484
$735$	31.0726	1.14613
$736$	−1.66948	−0.0615380
$737$	−28.0378	−1.03279
$738$	−7.93198	−0.291980
$739$	31.0761	1.14315	0.571577	−	0.820549i	$-0.306332\pi$
$739$	31.0761	1.14315	0.571577	+	0.820549i	$0.306332\pi$
$740$	15.6027	0.573565
$741$	0	0
$742$	−49.5798	−1.82013
$743$	33.6944	1.23613	0.618064	−	0.786128i	$-0.287917\pi$
$743$	33.6944	1.23613	0.618064	+	0.786128i	$0.287917\pi$
$744$	2.57842	0.0945294
$745$	−69.2456	−2.53696
$746$	−2.73685	−0.100203
$747$	0.911715	0.0333579
$748$	−11.0285	−0.403240
$749$	−43.5205	−1.59021
$750$	−4.79155	−0.174963
$751$	1.70343	0.0621589	0.0310794	−	0.999517i	$-0.490106\pi$
$751$	1.70343	0.0621589	0.0310794	+	0.999517i	$0.490106\pi$
$752$	−6.13466	−0.223708
$753$	5.53585	0.201738
$754$	0	0
$755$	−60.1895	−2.19052
$756$	−20.9850	−0.763216
$757$	35.2712	1.28195	0.640976	−	0.767561i	$-0.278530\pi$
$757$	35.2712	1.28195	0.640976	+	0.767561i	$0.278530\pi$
$758$	−1.62208	−0.0589167
$759$	−9.56218	−0.347085
$760$	2.98870	0.108411
$761$	6.63116	0.240379	0.120190	−	0.992751i	$-0.461650\pi$
$761$	6.63116	0.240379	0.120190	+	0.992751i	$0.461650\pi$
$762$	−16.9209	−0.612978
$763$	22.6523	0.820070
$764$	−20.8136	−0.753010
$765$	6.43958	0.232824
$766$	31.9773	1.15539
$767$	0	0
$768$	1.50159	0.0541841
$769$	−19.7122	−0.710840	−0.355420	−	0.934707i	$-0.615662\pi$
$769$	−19.7122	−0.710840	−0.355420	+	0.934707i	$0.615662\pi$
$770$	42.5386	1.53298
$771$	19.0826	0.687243
$772$	21.3064	0.766833
$773$	10.8204	0.389184	0.194592	−	0.980884i	$-0.437662\pi$
$773$	10.8204	0.389184	0.194592	+	0.980884i	$0.437662\pi$
$774$	−4.04444	−0.145374
$775$	6.75227	0.242549
$776$	14.8790	0.534124
$777$	29.2515	1.04939
$778$	−4.42183	−0.158530
$779$	10.6438	0.381355
$780$	0	0
$781$	−37.2281	−1.33213
$782$	4.82697	0.172612
$783$	47.8722	1.71081
$784$	6.92379	0.247278
$785$	−39.1221	−1.39633
$786$	−19.1628	−0.683515
$787$	−48.0124	−1.71146	−0.855729	−	0.517424i	$-0.826891\pi$
$787$	−48.0124	−1.71146	−0.855729	+	0.517424i	$0.826891\pi$
$788$	−8.62246	−0.307162
$789$	4.87891	0.173694
$790$	9.09981	0.323757
$791$	−19.7429	−0.701975
$792$	−2.84253	−0.101005
$793$	0	0
$794$	0.227725	0.00808165
$795$	−59.6294	−2.11484
$796$	4.00411	0.141922
$797$	22.8603	0.809755	0.404878	−	0.914371i	$-0.367314\pi$
$797$	22.8603	0.809755	0.404878	+	0.914371i	$0.367314\pi$
$798$	5.60313	0.198349
$799$	17.7371	0.627493
$800$	3.93232	0.139028
$801$	−11.0429	−0.390182
$802$	6.80150	0.240169
$803$	−0.450985	−0.0159149
$804$	−11.0376	−0.389266
$805$	−18.6184	−0.656214
$806$	0	0
$807$	16.4278	0.578285
$808$	14.4250	0.507469
$809$	16.3061	0.573294	0.286647	−	0.958036i	$-0.407459\pi$
$809$	16.3061	0.573294	0.286647	+	0.958036i	$0.407459\pi$
$810$	−18.5568	−0.652020
$811$	−0.810412	−0.0284574	−0.0142287	−	0.999899i	$-0.504529\pi$
$811$	−0.810412	−0.0284574	−0.0142287	+	0.999899i	$0.504529\pi$
$812$	−31.7638	−1.11469
$813$	−1.45437	−0.0510072
$814$	19.9131	0.697954
$815$	17.6409	0.617935
$816$	−4.34155	−0.151985
$817$	5.42719	0.189873
$818$	16.0814	0.562274
$819$	0	0
$820$	31.8112	1.11090
$821$	−6.51020	−0.227208	−0.113604	−	0.993526i	$-0.536239\pi$
$821$	−6.51020	−0.227208	−0.113604	+	0.993526i	$0.536239\pi$
$822$	−12.1688	−0.424434
$823$	22.1663	0.772670	0.386335	−	0.922359i	$-0.373741\pi$
$823$	22.1663	0.772670	0.386335	+	0.922359i	$0.373741\pi$
$824$	9.25772	0.322508
$825$	22.5229	0.784145
$826$	20.6212	0.717504
$827$	8.74542	0.304108	0.152054	−	0.988372i	$-0.451411\pi$
$827$	8.74542	0.304108	0.152054	+	0.988372i	$0.451411\pi$
$828$	1.24413	0.0432365
$829$	−22.2016	−0.771092	−0.385546	−	0.922689i	$-0.625987\pi$
$829$	−22.2016	−0.771092	−0.385546	+	0.922689i	$0.625987\pi$
$830$	−3.65644	−0.126917
$831$	12.0524	0.418093
$832$	0	0
$833$	−20.0187	−0.693607
$834$	14.8659	0.514764
$835$	49.7819	1.72278
$836$	3.81437	0.131923
$837$	−9.65674	−0.333786
$838$	−21.3995	−0.739234
$839$	31.7739	1.09696	0.548479	−	0.836164i	$-0.315207\pi$
$839$	31.7739	1.09696	0.548479	+	0.836164i	$0.315207\pi$
$840$	16.7461	0.577795
$841$	43.4616	1.49868
$842$	26.3895	0.909444
$843$	−6.63391	−0.228484
$844$	8.29916	0.285669
$845$	0	0
$846$	4.57165	0.157177
$847$	13.2444	0.455083
$848$	−13.2870	−0.456276
$849$	−23.7102	−0.813733
$850$	−11.3695	−0.389970
$851$	−8.71563	−0.298768
$852$	−14.6555	−0.502090
$853$	56.4010	1.93113	0.965567	−	0.260154i	$-0.0837733\pi$
$853$	56.4010	1.93113	0.965567	+	0.260154i	$0.0837733\pi$
$854$	−22.3571	−0.765046
$855$	−2.22723	−0.0761697
$856$	−11.6631	−0.398638
$857$	−26.8812	−0.918244	−0.459122	−	0.888373i	$-0.651836\pi$
$857$	−26.8812	−0.918244	−0.459122	+	0.888373i	$0.651836\pi$
$858$	0	0
$859$	21.5548	0.735440	0.367720	−	0.929937i	$-0.380138\pi$
$859$	21.5548	0.735440	0.367720	+	0.929937i	$0.380138\pi$
$860$	16.2202	0.553106
$861$	59.6389	2.03249
$862$	22.5479	0.767984
$863$	−28.9587	−0.985767	−0.492883	−	0.870095i	$-0.664057\pi$
$863$	−28.9587	−0.985767	−0.492883	+	0.870095i	$0.664057\pi$
$864$	−5.62379	−0.191325
$865$	−28.3116	−0.962624
$866$	11.5129	0.391223
$867$	−12.9744	−0.440633
$868$	6.40737	0.217480
$869$	11.6137	0.393969
$870$	−38.2022	−1.29518
$871$	0	0
$872$	6.07064	0.205578
$873$	−11.0881	−0.375275
$874$	−1.66948	−0.0564711
$875$	−11.9070	−0.402530
$876$	−0.177538	−0.00599846
$877$	9.29627	0.313913	0.156956	−	0.987606i	$-0.449832\pi$
$877$	9.29627	0.313913	0.156956	+	0.987606i	$0.449832\pi$
$878$	−11.6448	−0.392994
$879$	51.0629	1.72231
$880$	11.4000	0.384294
$881$	−16.6824	−0.562046	−0.281023	−	0.959701i	$-0.590674\pi$
$881$	−16.6824	−0.562046	−0.281023	+	0.959701i	$0.590674\pi$
$882$	−5.15973	−0.173737
$883$	44.8252	1.50849	0.754245	−	0.656594i	$-0.228003\pi$
$883$	44.8252	1.50849	0.754245	+	0.656594i	$0.228003\pi$
$884$	0	0
$885$	24.8011	0.833678
$886$	26.0168	0.874050
$887$	51.4991	1.72917	0.864586	−	0.502485i	$-0.167581\pi$
$887$	51.4991	1.72917	0.864586	+	0.502485i	$0.167581\pi$
$888$	7.83915	0.263065
$889$	−42.0483	−1.41026
$890$	44.2877	1.48453
$891$	−23.6834	−0.793423
$892$	8.81599	0.295181
$893$	−6.13466	−0.205288
$894$	−34.7906	−1.16357
$895$	−14.9567	−0.499946
$896$	3.73146	0.124659
$897$	0	0
$898$	11.7314	0.391482
$899$	−14.6169	−0.487501
$900$	−2.93043	−0.0976812
$901$	38.4165	1.27984
$902$	40.5995	1.35182
$903$	30.4093	1.01196
$904$	−5.29092	−0.175973
$905$	−5.89967	−0.196112
$906$	−30.2406	−1.00468
$907$	31.7073	1.05282	0.526411	−	0.850230i	$-0.323537\pi$
$907$	31.7073	1.05282	0.526411	+	0.850230i	$0.323537\pi$
$908$	−4.18895	−0.139015
$909$	−10.7497	−0.356546
$910$	0	0
$911$	−46.0327	−1.52513	−0.762566	−	0.646911i	$-0.776061\pi$
$911$	−46.0327	−1.52513	−0.762566	+	0.646911i	$0.776061\pi$
$912$	1.50159	0.0497227
$913$	−4.66657	−0.154441
$914$	−36.7678	−1.21617
$915$	−26.8888	−0.888918
$916$	−24.2635	−0.801689
$917$	−47.6196	−1.57254
$918$	16.2600	0.536661
$919$	16.5907	0.547278	0.273639	−	0.961832i	$-0.411773\pi$
$919$	16.5907	0.547278	0.273639	+	0.961832i	$0.411773\pi$
$920$	−4.98959	−0.164502
$921$	−43.8700	−1.44557
$922$	−12.0022	−0.395270
$923$	0	0
$924$	21.3724	0.703100
$925$	20.5289	0.674985
$926$	15.5101	0.509693
$927$	−6.89902	−0.226593
$928$	−8.51244	−0.279434
$929$	−48.8958	−1.60422	−0.802110	−	0.597176i	$-0.796289\pi$
$929$	−48.8958	−1.60422	−0.802110	+	0.597176i	$0.796289\pi$
$930$	7.70612	0.252693
$931$	6.92379	0.226918
$932$	17.9212	0.587027
$933$	−19.7484	−0.646534
$934$	−25.8302	−0.845189
$935$	−32.9607	−1.07793
$936$	0	0
$937$	23.3908	0.764144	0.382072	−	0.924133i	$-0.375211\pi$
$937$	23.3908	0.764144	0.382072	+	0.924133i	$0.375211\pi$
$938$	−27.4284	−0.895570
$939$	33.3096	1.08702
$940$	−18.3346	−0.598010
$941$	17.0539	0.555940	0.277970	−	0.960590i	$-0.410338\pi$
$941$	17.0539	0.555940	0.277970	+	0.960590i	$0.410338\pi$
$942$	−19.6559	−0.640423
$943$	−17.7697	−0.578662
$944$	5.52632	0.179866
$945$	−62.7177	−2.04021
$946$	20.7013	0.673057
$947$	−17.9107	−0.582021	−0.291010	−	0.956720i	$-0.593991\pi$
$947$	−17.9107	−0.582021	−0.291010	+	0.956720i	$0.593991\pi$
$948$	4.57196	0.148490
$949$	0	0
$950$	3.93232	0.127581
$951$	−37.9800	−1.23159
$952$	−10.7887	−0.349665
$953$	8.02689	0.260016	0.130008	−	0.991513i	$-0.458500\pi$
$953$	8.02689	0.260016	0.130008	+	0.991513i	$0.458500\pi$
$954$	9.90169	0.320579
$955$	−62.2056	−2.01293
$956$	−22.3178	−0.721808
$957$	−48.7560	−1.57606
$958$	10.0622	0.325094
$959$	−30.2394	−0.976481
$960$	4.48781	0.144843
$961$	−28.0515	−0.904887
$962$	0	0
$963$	8.69158	0.280082
$964$	22.3962	0.721334
$965$	63.6783	2.04988
$966$	−9.35434	−0.300971
$967$	31.4562	1.01156	0.505781	−	0.862662i	$-0.331204\pi$
$967$	31.4562	1.01156	0.505781	+	0.862662i	$0.331204\pi$
$968$	3.54939	0.114082
$969$	−4.34155	−0.139471
$970$	44.4688	1.42781
$971$	−16.7191	−0.536542	−0.268271	−	0.963343i	$-0.586452\pi$
$971$	−16.7191	−0.536542	−0.268271	+	0.963343i	$0.586452\pi$
$972$	7.54799	0.242102
$973$	36.9418	1.18430
$974$	−5.76723	−0.184794
$975$	0	0
$976$	−5.99153	−0.191784
$977$	−9.07353	−0.290288	−0.145144	−	0.989411i	$-0.546365\pi$
$977$	−9.07353	−0.290288	−0.145144	+	0.989411i	$0.546365\pi$
$978$	8.86323	0.283415
$979$	56.5227	1.80647
$980$	20.6931	0.661017
$981$	−4.52395	−0.144439
$982$	19.4673	0.621226
$983$	48.1584	1.53601	0.768007	−	0.640442i	$-0.221249\pi$
$983$	48.1584	1.53601	0.768007	+	0.640442i	$0.221249\pi$
$984$	15.9827	0.509511
$985$	−25.7699	−0.821098
$986$	24.6120	0.783804
$987$	−34.3733	−1.09411
$988$	0	0
$989$	−9.06061	−0.288111
$990$	−8.49548	−0.270004
$991$	−43.3084	−1.37574	−0.687869	−	0.725835i	$-0.741454\pi$
$991$	−43.3084	−1.37574	−0.687869	+	0.725835i	$0.741454\pi$
$992$	1.71712	0.0545187
$993$	41.4793	1.31630
$994$	−36.4190	−1.15514
$995$	11.9671	0.379382
$996$	−1.83708	−0.0582101
$997$	−57.6729	−1.82652	−0.913260	−	0.407377i	$-0.866444\pi$
$997$	−57.6729	−1.82652	−0.913260	+	0.407377i	$0.866444\pi$
$998$	4.74563	0.150220
$999$	−29.3593	−0.928888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bn.1.9		14
13.6	odd	12		494.2.m.b.153.3	✓	28
13.11	odd	12		494.2.m.b.381.3	yes	28
13.12	even	2		6422.2.a.bm.1.9		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.m.b.153.3	✓	28	13.6	odd	12
494.2.m.b.381.3	yes	28	13.11	odd	12
6422.2.a.bm.1.9		14	13.12	even	2
6422.2.a.bn.1.9		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.50159 q^{3} +1.00000 q^{4} +2.98870 q^{5} +1.50159 q^{6} +3.73146 q^{7} +1.00000 q^{8} -0.745218 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.50159 q^{3} +1.00000 q^{4} +2.98870 q^{5} +1.50159 q^{6} +3.73146 q^{7} +1.00000 q^{8} -0.745218 q^{9} +2.98870 q^{10} +3.81437 q^{11} +1.50159 q^{12} +3.73146 q^{14} +4.48781 q^{15} +1.00000 q^{16} -2.89129 q^{17} -0.745218 q^{18} +1.00000 q^{19} +2.98870 q^{20} +5.60313 q^{21} +3.81437 q^{22} -1.66948 q^{23} +1.50159 q^{24} +3.93232 q^{25} -5.62379 q^{27} +3.73146 q^{28} -8.51244 q^{29} +4.48781 q^{30} +1.71712 q^{31} +1.00000 q^{32} +5.72763 q^{33} -2.89129 q^{34} +11.1522 q^{35} -0.745218 q^{36} +5.22055 q^{37} +1.00000 q^{38} +2.98870 q^{40} +10.6438 q^{41} +5.60313 q^{42} +5.42719 q^{43} +3.81437 q^{44} -2.22723 q^{45} -1.66948 q^{46} -6.13466 q^{47} +1.50159 q^{48} +6.92379 q^{49} +3.93232 q^{50} -4.34155 q^{51} -13.2870 q^{53} -5.62379 q^{54} +11.4000 q^{55} +3.73146 q^{56} +1.50159 q^{57} -8.51244 q^{58} +5.52632 q^{59} +4.48781 q^{60} -5.99153 q^{61} +1.71712 q^{62} -2.78075 q^{63} +1.00000 q^{64} +5.72763 q^{66} -7.35059 q^{67} -2.89129 q^{68} -2.50689 q^{69} +11.1522 q^{70} -9.75998 q^{71} -0.745218 q^{72} -0.118233 q^{73} +5.22055 q^{74} +5.90474 q^{75} +1.00000 q^{76} +14.2332 q^{77} +3.04474 q^{79} +2.98870 q^{80} -6.20900 q^{81} +10.6438 q^{82} -1.22342 q^{83} +5.60313 q^{84} -8.64121 q^{85} +5.42719 q^{86} -12.7822 q^{87} +3.81437 q^{88} +14.8184 q^{89} -2.22723 q^{90} -1.66948 q^{92} +2.57842 q^{93} -6.13466 q^{94} +2.98870 q^{95} +1.50159 q^{96} +14.8790 q^{97} +6.92379 q^{98} -2.84253 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9}+O(q^{10}) \) 14 * q + 14 * q^2 + 4 * q^3 + 14 * q^4 - 2 * q^5 + 4 * q^6 + 2 * q^7 + 14 * q^8 + 18 * q^9	\( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9} - 2 q^{10} + 10 q^{11} + 4 q^{12} + 2 q^{14} - 4 q^{15} + 14 q^{16} + 2 q^{17} + 18 q^{18} + 14 q^{19} - 2 q^{20} - 18 q^{21} + 10 q^{22} + 12 q^{23} + 4 q^{24} + 44 q^{25} + 10 q^{27} + 2 q^{28} + 4 q^{29} - 4 q^{30} + 4 q^{31} + 14 q^{32} + 12 q^{33} + 2 q^{34} + 14 q^{35} + 18 q^{36} + 18 q^{37} + 14 q^{38} - 2 q^{40} + 6 q^{41} - 18 q^{42} + 28 q^{43} + 10 q^{44} + 8 q^{45} + 12 q^{46} - 20 q^{47} + 4 q^{48} + 28 q^{49} + 44 q^{50} + 10 q^{51} + 12 q^{53} + 10 q^{54} - 2 q^{55} + 2 q^{56} + 4 q^{57} + 4 q^{58} + 16 q^{59} - 4 q^{60} + 30 q^{61} + 4 q^{62} + 28 q^{63} + 14 q^{64} + 12 q^{66} + 2 q^{67} + 2 q^{68} - 42 q^{69} + 14 q^{70} + 44 q^{71} + 18 q^{72} - 36 q^{73} + 18 q^{74} + 46 q^{75} + 14 q^{76} + 68 q^{77} + 34 q^{79} - 2 q^{80} - 6 q^{81} + 6 q^{82} + 2 q^{83} - 18 q^{84} - 30 q^{85} + 28 q^{86} + 52 q^{87} + 10 q^{88} + 42 q^{89} + 8 q^{90} + 12 q^{92} - 12 q^{93} - 20 q^{94} - 2 q^{95} + 4 q^{96} + 40 q^{97} + 28 q^{98} + 12 q^{99}+O(q^{100}) \) 14 * q + 14 * q^2 + 4 * q^3 + 14 * q^4 - 2 * q^5 + 4 * q^6 + 2 * q^7 + 14 * q^8 + 18 * q^9 - 2 * q^10 + 10 * q^11 + 4 * q^12 + 2 * q^14 - 4 * q^15 + 14 * q^16 + 2 * q^17 + 18 * q^18 + 14 * q^19 - 2 * q^20 - 18 * q^21 + 10 * q^22 + 12 * q^23 + 4 * q^24 + 44 * q^25 + 10 * q^27 + 2 * q^28 + 4 * q^29 - 4 * q^30 + 4 * q^31 + 14 * q^32 + 12 * q^33 + 2 * q^34 + 14 * q^35 + 18 * q^36 + 18 * q^37 + 14 * q^38 - 2 * q^40 + 6 * q^41 - 18 * q^42 + 28 * q^43 + 10 * q^44 + 8 * q^45 + 12 * q^46 - 20 * q^47 + 4 * q^48 + 28 * q^49 + 44 * q^50 + 10 * q^51 + 12 * q^53 + 10 * q^54 - 2 * q^55 + 2 * q^56 + 4 * q^57 + 4 * q^58 + 16 * q^59 - 4 * q^60 + 30 * q^61 + 4 * q^62 + 28 * q^63 + 14 * q^64 + 12 * q^66 + 2 * q^67 + 2 * q^68 - 42 * q^69 + 14 * q^70 + 44 * q^71 + 18 * q^72 - 36 * q^73 + 18 * q^74 + 46 * q^75 + 14 * q^76 + 68 * q^77 + 34 * q^79 - 2 * q^80 - 6 * q^81 + 6 * q^82 + 2 * q^83 - 18 * q^84 - 30 * q^85 + 28 * q^86 + 52 * q^87 + 10 * q^88 + 42 * q^89 + 8 * q^90 + 12 * q^92 - 12 * q^93 - 20 * q^94 - 2 * q^95 + 4 * q^96 + 40 * q^97 + 28 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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